STAT 17: Statistical Methods for Business and Economics

Prof. Marcela Alfaro Cordoba

Statistics - UCSC

28 Apr 2026

Meet Sofia: Fighting Climate Change 🌍

Sofia is a UCSC environmental studies student working with a campus sustainability organization…

Her mission: Increase student participation in climate action programs

  • Bike-sharing program
  • Zero-waste dining initiatives
  • Community gardens
  • Climate justice workshops

The Challenge: With limited resources, how can she predict which students are most likely to participate? 🤔

Today’s goal: Use probability to make strategic decisions about outreach!

What We’ll Accomplish Today

By the end of this lecture, you will be able to:

  • ✅ Apply fundamental probability terminology correctly (experiment, outcome, event, sample space)
  • ✅ Distinguish between independent and mutually exclusive events
  • ✅ Apply the addition rule for probability calculations
  • ✅ Apply the multiplication rule for probability calculations

What is Probability? 🎯

Probability measures the likelihood that an event will occur.

Scale: Always between 0 and 1 (or 0% to 100%)

  • P = 0: Event is impossible
  • P = 0.5: Event is equally likely to occur or not
  • P = 1: Event is certain

Sofia’s question: “What’s the probability a randomly selected UCSC student will join our bike-sharing program?”

Why Probability Matters 💡

In real life, we use probability constantly:

  • Weather forecasts: “70% chance of rain tomorrow”
  • Medical decisions: “This treatment has an 85% success rate”
  • Sports predictions: “The team has a 60% chance of winning”
  • Education: “Students who attend office hours have a 40% higher pass rate”
  • Climate action: “Students living on campus are twice as likely to use bike-sharing”

Probability helps us make informed decisions under uncertainty! 🎲

Fundamental Probability Terminology 📚

Experiment
Any process that generates observations or outcomes
Outcome
A single result of an experiment
Sample Space (S)
The set of all possible outcomes
Event (E)
A collection of one or more outcomes (subset of sample space)

Example: Random Student Survey

Sofia’s Experiment: Select one random UCSC student and ask about their climate action participation.

Possible Outcomes:

  • Uses bike-sharing
  • Participates in zero-waste dining
  • Volunteers in community garden
  • Attends climate workshops
  • None of the above

Sample Space (S): {Bike-sharing, Zero-waste, Garden, Workshops, None}

Event E: “Student participates in at least one program”

E = {Bike-sharing, Zero-waste, Garden, Workshops}

Classical Probability Definition

When all outcomes are equally likely:

\[P(E) = \frac{\text{Number of outcomes in } E}{\text{Total number of outcomes in } S}\]

Example: Roll a fair six-sided die

  • Sample Space: S = {1, 2, 3, 4, 5, 6}
  • Event E: “Roll an even number” = {2, 4, 6}
  • P(E) = 3/6 = 0.5 or 50%

Empirical Probability (Real Data!)

When outcomes are NOT equally likely, we use observed data:

\[P(E) = \frac{\text{Number of times } E \text{ occurred}}{\text{Total number of trials}}\]

Sofia’s real data: She surveyed 500 UCSC students

  • 125 use bike-sharing
  • 200 participate in zero-waste dining
  • 75 volunteer in community garden
  • 100 attend workshops

P(Uses bike-sharing) = 125/500 = 0.25 or 25%

Calculating Probabilities from Data

Sofia’s survey results (n = 500 students):

Program Count Probability
Bike-sharing 125 125/500 = 0.25
Zero-waste dining 200 200/500 = 0.40
Community garden 75 75/500 = 0.15
Climate workshops 100 100/500 = 0.20

Question: What’s the probability a random student does NOT use bike-sharing?

Answer: P(No bike-sharing) = 1 - 0.25 = 0.75 or 75%

The Complement Rule 🔄

Complement of Event E (denoted Ec or E’):
All outcomes in the sample space that are NOT in E

Complement Rule:

\[P(E^c) = 1 - P(E)\]

Why? The total probability must equal 1:

\[P(E) + P(E^c) = 1\]

Complement Rule Example

Sofia’s data: P(Participates in zero-waste) = 0.40

Find: P(Does NOT participate in zero-waste)

Solution:

P(Does NOT participate) = 1 - P(Participates)

P(Does NOT participate) = 1 - 0.40 = 0.60 or 60%

Interpretation: 60% of students are not currently participating in zero-waste dining - a potential target for outreach! 🎯

📊 THINK-PAIR-SHARE #1 (5 minutes)

Probability Practice with Real Data:

Sofia surveyed 400 students about their primary transportation to campus:

  • Walk: 160 students
  • Bike: 100 students
  • Bus: 80 students
  • Car: 60 students

Calculate:

  1. P(Student bikes to campus)
  2. P(Student does NOT bike to campus)
  3. P(Student uses active transportation - walk or bike)

Discuss with a partner: How could Sofia use these probabilities to plan bike infrastructure improvements?

Share your answers in Poll Everywhere!

  1. P(Student bikes to campus)
  2. P(Student does NOT bike to campus)
  3. P(Student uses active transportation - walk or bike)

Types of Events: Introduction 🎲

Sofia needs to understand how events relate to each other:

Two critical concepts:

  1. Mutually Exclusive Events - Cannot happen at the same time
  2. Independent Events - One event doesn’t affect the other

Important: These are DIFFERENT concepts! Many students confuse them! ⚠️

Mutually Exclusive Events 🚫

Definition: Two events are mutually exclusive (or disjoint) if they cannot occur at the same time.

\[\text{If } A \text{ and } B \text{ are mutually exclusive: } P(A \text{ and } B) = 0\]

Key insight: If one happens, the other CANNOT happen!

Mutually Exclusive: Examples

Example 1: Student’s primary residence

Event A: “Lives on campus”
Event B: “Lives off campus”

Are they mutually exclusive? YES! ✅

A student cannot live both on AND off campus simultaneously.

Example 2: Student’s participation

Event C: “Uses bike-sharing”
Event D: “Participates in zero-waste dining”

Are they mutually exclusive? NO! ❌

A student can do BOTH! These events can overlap.

Visualizing Mutually Exclusive Events

Venn Diagram - Mutually Exclusive:

(polygon[GRID.polygon.1], polygon[GRID.polygon.2], polygon[GRID.polygon.3], polygon[GRID.polygon.4], text[GRID.text.5], text[GRID.text.6], text[GRID.text.7], text[GRID.text.8])

No overlap = Mutually exclusive!

Independent Events 🎯

Definition: Two events are independent if the occurrence of one event does NOT affect the probability of the other event.

\[\text{If } A \text{ and } B \text{ are independent: } P(A \text{ and } B) = P(A) \times P(B)\]

Key insight: Knowing one happened tells you NOTHING about whether the other happened!

Independent: Examples

Example 1: Two different students

Event A: “Student 1 uses bike-sharing”
Event B: “Student 2 uses bike-sharing”

Are they independent? YES! ✅

What Student 1 does doesn’t affect what Student 2 does.

Example 2: Same student, related behaviors

Event C: “Student lives on campus”
Event D: “Student uses bike-sharing”

Are they independent? Probably NO! ❌

Students living on campus might be MORE likely to bike (campus is closer). These events are likely dependent.

Mutually Exclusive vs Independent ⚠️

CRITICAL DISTINCTION:

Mutually Exclusive

  • Events cannot occur together
  • If A happens, B cannot
  • P(A and B) = 0
  • About overlap

Example: “Lives on campus” and “Lives off campus”

Independent

  • Events do not affect each other
  • A happening doesn’t change P(B)
  • P(A and B) = P(A) × P(B)
  • About influence

Example: Flipping a coin twice - first flip doesn’t affect second

Common Misconception! 🚨

False statement: “If events are mutually exclusive, they must be independent.”

Truth: If events are mutually exclusive, they are DEPENDENT!

Why? If you know A happened, you KNOW B did not happen!

  • Knowing A occurred gives you information about B
  • Therefore, they are dependent

Exception: If P(A) = 0 or P(B) = 0 (impossible events)

🧘‍♀️ STRETCH BREAK

Time to move! (5 minutes)

  • Stand up and stretch 🤸‍♀️
  • Chat with neighbors 💬
  • Grab some water 💧

The Addition Rule 📊

Question: What’s the probability that at least one of two events occurs?

General Addition Rule:

\[P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\]

Why subtract P(A and B)?

To avoid counting the overlap twice! 🎯

Addition Rule: Mutually Exclusive Case

If A and B are mutually exclusive: P(A and B) = 0

Simplified Addition Rule:

\[P(A \text{ or } B) = P(A) + P(B)\]

Example: Student’s primary residence

  • P(Lives on campus) = 0.65
  • P(Lives off campus) = 0.35
  • P(Lives on campus OR off campus) = 0.65 + 0.35 = 1.00 ✅

Makes sense! Every student lives somewhere.

Addition Rule: General Case

Example: Sofia’s survey (n = 500)

  • 125 use bike-sharing
  • 200 participate in zero-waste dining
  • 50 do BOTH

Find: P(Student uses bike-sharing OR participates in zero-waste)

Solution:

P(Bike OR Zero-waste) = P(Bike) + P(Zero-waste) - P(Both)

= 125/500 + 200/500 - 50/500

= 0.25 + 0.40 - 0.10

= 0.55 or 55%

Visualizing the Addition Rule

Venn Diagram - With Overlap:

(polygon[GRID.polygon.9], polygon[GRID.polygon.10], polygon[GRID.polygon.11], polygon[GRID.polygon.12], text[GRID.text.13], text[GRID.text.14], text[GRID.text.15], text[GRID.text.16], text[GRID.text.17])

Green + Blue + Overlap (counted once) = Total students doing at least one!

Addition Rule Practice

Sofia’s expanded data (n = 800 students):

  • 240 use bike-sharing: P(B) = 240/800 = 0.30
  • 320 participate in zero-waste: P(Z) = 320/800 = 0.40
  • 80 do both: P(B and Z) = 80/800 = 0.10

Verify with Addition Rule:

P(B or Z) = 0.30 + 0.40 - 0.10 = 0.60

Check: 240 + 320 - 80 = 480 students

480/800 = 0.60 ✅

The Multiplication Rule 🎲

Question: What’s the probability that both of two events occur?

General Multiplication Rule:

\[P(A \text{ and } B) = P(A) \times P(B|A)\]

Where P(B|A) means “probability of B given that A occurred”

For independent events, this simplifies!

Multiplication Rule: Independent Case

If A and B are independent: P(B|A) = P(B)

Simplified Multiplication Rule:

\[P(A \text{ and } B) = P(A) \times P(B)\]

Example: Two random students

  • P(Student 1 uses bike-sharing) = 0.25
  • P(Student 2 uses bike-sharing) = 0.25

P(Both use bike-sharing) = 0.25 × 0.25 = 0.0625 or 6.25%

Multiplication Rule: Multiple Events

For three independent events:

\[P(A \text{ and } B \text{ and } C) = P(A) \times P(B) \times P(C)\]

Example: Three random students

P(All three use bike-sharing) = 0.25 × 0.25 × 0.25 = 0.0156 or 1.56%

Notice: As we add more independent events, the probability of ALL occurring gets smaller! 📉

Multiplication Rule Application

Sofia’s outreach strategy:

She sends emails to students. From past data:

  • P(Student opens email) = 0.40
  • P(Student clicks link | opened email) = 0.30
  • These are independent? NO! Must have opened to click.

Find: P(Student opens email AND clicks link)

Solution:

P(Opens and Clicks) = P(Opens) × P(Clicks | Opens)

= 0.40 × 0.30 = 0.12 or 12%

When to Use Which Rule? 🤔

Addition Rule

Use when finding: “A OR B”

  • P(at least one occurs)
  • P(either one happens)

Formula:

P(A or B) = P(A) + P(B) - P(A and B)

If mutually exclusive:

P(A or B) = P(A) + P(B)

Multiplication Rule

Use when finding: “A AND B”

  • P(both occur)
  • P(all happen together)

Formula:

P(A and B) = P(A) × P(B|A)

If independent:

P(A and B) = P(A) × P(B)

📊 THINK-PAIR-SHARE #2 (7 minutes)

Real-World Probability Challenge:

Sofia’s climate action survey (n = 600):

  • 180 students are first-generation college students
  • 240 students live on campus
  • 90 students are BOTH first-gen AND live on campus

Tasks:

  1. Calculate P(First-gen OR Lives on campus) using the addition rule
  2. Are “first-gen” and “lives on campus” mutually exclusive? Why or why not?
  3. If these events were independent, what would P(First-gen AND Lives on campus) be? Compare to the actual value (90/600 = 0.15).
  4. What does this tell Sofia about targeting her outreach?

Post your analysis on Ed Discussion with your group members’ names!

Real Applications for Sofia 🌱

Using probability to improve outreach:

Finding 1: P(Uses bike-sharing | Lives on campus) = 0.35

P(Uses bike-sharing | Lives off campus) = 0.12

Action: Focus bike infrastructure near dorms! 🚲

Finding 2: P(Attends workshop AND volunteers in garden) = 0.05

P(Attends workshop) × P(Garden volunteer) = 0.20 × 0.15 = 0.03

Insight: These activities are NOT independent - students who do one are more likely to do the other! Create combined programs! 🌿

Summary: Probability Rules

Key Formulas:

Complement Rule: \[P(E^c) = 1 - P(E)\]

Addition Rule (for A or B): \[P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\]

If mutually exclusive: P(A or B) = P(A) + P(B)

Multiplication Rule (for A and B): \[P(A \text{ and } B) = P(A) \times P(B|A)\]

If independent: P(A and B) = P(A) × P(B)

Calculating Probabilities in Google Sheets 💻

Example dataset in columns A-C:

Student ID Uses Bike Zero-Waste
1 Yes Yes
2 No Yes
… … …

Count events:

  • =COUNTIF(B:B, "Yes") - counts bike users
  • =COUNTIFS(B:B, "Yes", C:C, "Yes") - counts both

Calculate probability:

  • =COUNTIF(B:B, "Yes")/COUNTA(B:B) - P(Bike)

Decision Trees for Complex Probabilities 🌳

Visual tool for sequential events:

Common Probability Mistakes ⚠️

  1. Confusing mutually exclusive with independent - they’re opposite concepts!
  2. Forgetting to subtract overlap in addition rule when events aren’t mutually exclusive
  3. Assuming independence when events are related (must verify!)
  4. Adding probabilities when you should multiply (mixing up AND vs OR)
  5. Probabilities > 1 - always check! Probabilities must be between 0 and 1

Sofia’s Success Story! 🎉

How probability transformed her outreach:

✅ Targeted campaigns: Focused on on-campus students for bike-sharing (35% participation vs 12% off-campus)

✅ Bundle programs: Created combined workshop + garden volunteer events (since they’re not independent!)

✅ Realistic goals: Expected 12% email click rate instead of overestimating

✅ Impact: Increased overall climate action participation by 28% in one semester! 🌱

Your Probability Toolkit 🧰

Fundamental Concepts:

  • Sample space = all possible outcomes
  • Event = subset of sample space
  • P(E) = number of favorable outcomes / total outcomes
  • Complement: P(Ec) = 1 - P(E)

Event Types:

  • Mutually exclusive: Cannot happen together, P(A and B) = 0
  • Independent: Don’t affect each other, P(A and B) = P(A) × P(B)

Calculation Rules:

  • Addition (OR): P(A or B) = P(A) + P(B) - P(A and B)
  • Multiplication (AND): P(A and B) = P(A) × P(B) if independent

Looking Ahead

Next Class: Conditional Probability and Contingency Tables

  • Understanding P(A|B)
  • Contingency tables and simple, marginal and joint probability

This week’s assignment: Analyze probability scenarios using real campus data! 📊

Questions? Probability can be tricky - office hours are here to help! 🤝

Quick Knowledge Check ✅

Rate your confidence (1-5) on Ed Discussion:

  1. Applying probability terminology correctly ⭐⭐⭐⭐⭐
  2. Distinguishing between mutually exclusive and independent events ⭐⭐⭐⭐⭐
  3. Using the addition rule correctly ⭐⭐⭐⭐⭐
  4. Using the multiplication rule correctly ⭐⭐⭐⭐⭐
  5. Solving real-world probability problems ⭐⭐⭐⭐⭐

If you rated anything 3 or below, please visit office hours! 🤗

Thank you! 🎲✨

Questions? Office hours information on Canvas.

Next up: Conditional Probability & Bayes’ Theorem!